Connectness of quaternion

Question

How can I prove that the set of all non-zero quaternions is connected?

I'm not sure how we can have subsets of quaternions, as for real number, we can just use intervals, but we can't do that for quaternions.

Answer 1

Hint: the nonzero quaternions are path connected, either because the straight line joining two of them doesn't pass through zero, or because one could take a 'side-tour' so as to not walk through the point zero.